and so the singleton has measure zero. But is countably additive, so given any countable collection the measure is the sum of the measures of the individual points, each of which is zero.

Next, as I said when I introduced semiclosed intervals, we could have started with open intervals, but the details would have been messier. Now we can see that the -ring generated by the collection of semiclosed intervals is the same as that generated by the collection of all open sets.

We can see, in particular, that each open interval is a Borel set. Indeed, the point is a Borel set, as is the semiclosed interval , and we have the relation . Every other open set in is a countable union of open intervals, and so they’re all Borel sets as well. Conversely, we could write

and find the singleton in the -ring generated by . Then we can write and find every semiclosed interval in this -ring as well. And thus

We can also tie our current measure back to the concept of outer Lebesgue measure we introduced before. Back then, we defined the “volume” of a collection of open intervals to be the sum of the “volumes” of the intervals themselves. We defined the outer measure of a set to be the infimum of the volumes of finite open covers. And, indeed, this is exactly the outer measure corresponding to Lebesgue measure .

Remember that the outer measure is defined for a set by

Since , we have the inequality

On the other hand, if is any positive number, then by the definition of we can find a sequence of semiclosed intervals so that

and

We can thus widen each of these semiclosed intervals just a bit to find

and

Since was arbitrary, we find that . And, thus, that

In effect, we’ve replaced the messily-defined “volume” of an open cover by the more precise Lebesgue measure , but the result is the same. The “outer Lebesgue measure” from our investigations of multiple integrals is the same as the outer measure induced by our new Lebesgue measure.

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I’m still reading and pondering these with delight. Haven’t commented much lately, as had to complete a grant proposal, make progress in the quantum computing theory in a novel I’m writing, and be the key eyewitness in a legal malpractice lawsuit. And — for complicated job-search reasons — formally applied to the Caltech Math department, where I got my B.S. in 1973, for Grad School next year (albeit I’ve been an adjunct professor in the interim and published a great deal). Your unapologetic blogmaster and some of his readers are also sensitive to the job search mess during Global Recession.

[…] . This sends the semiclosed interval to the interval . But this is a Borel set: , and the singletons are Borel sets. Thus we see that the reflection sends Borel sets to Borel sets. It should also be clear that it […]

[…] binary expansions, which are exactly the rational numbers. But this is a countable set, and countable sets have Lebesgue measure zero. Consequently, we find that . Since , there must be some positive measure in in order to make up […]

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This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.